Conventionally, lubricating oils are used for smoothing the operation of internal combustion engines, transmissions and other mechanical devices. In particular, lubricating oils for internal combustion engines (engine oils) are required to be high-performance as the internal combustion engines are designed to provide higher performances and higher powers, and be operated under increasingly severe conditions. Accordingly, in order to meet such performances required, various additives such as anti-wear agents, metallic detergents, ashless dispersants and antioxidants are used for conventional engine oils (see, for example, Patent documents 1 to 3). Recently, as the fuel saving performance required for lubricating oil is getting higher, considerations have been given to applications of high viscosity index base oil and various friction modifiers (see, for example, Patent document 4).    [Patent document 1] Japanese Unexamined Patent Publication No. 2001-279287    [Patent document 2] Japanese Unexamined Patent Publication No. 2002-129182    [Patent document 3] Japanese Unexamined Patent Publication HEI No. 08-302378    [Patent document 4] Japanese Unexamined Patent Publication HEI No. 06-306384